Submodular separation conjecture for low-degree monomial families

Let $0\leqslant r\leqslant d$ be integers and let $n=\binom{d}{\leqslant r}$. Define

$$\phi:\{0,1\}^d\to\{0,1\}^n, \qquad \phi(x)=\left(\prod_{i\in S}x_i\right)_{S\subset[d]:|S|\leqslant r},$$

and let ${\mathcal F}_{r,d}=\phi(\{0,1\}^d)$. A function $f:\{0,1\}^n\to\mathbb Z$ is submodular if $f(x)+f(y)\geqslant f(x\vee y)+f(x\wedge y)$ for all $x,y$.

Submodular separation conjecture. For every such $r,d$, there exists a submodular function $f$ such that $f(x)$ is computable in $\mathrm{poly}(n)$ time, $f(x)\leqslant-1$ for every $x\in\mathcal F_{r,d}$, and $f(x)\geqslant0$ for every $x\notin\mathcal F_{r,d}$. If true, this would yield a hardness result for submodular minimization with modular constraints, complementing the known prime-power algorithms; the source does not resolve the conjecture.